Perfectoid universal covers of curves

In a previous post I mumbled a bit about perfectoid covering spaces of rigid analytic varieties.  In this post, I want to sketch a fun special case.

Fix {K/\mathbf{Q}_p} an algebraically closed nonarchimedean field. Recall the following from Scholze-Weinstein: given a cofiltered inverse system {(X_i)_{i\in I}} of rigid spaces over {K} with qcqs transition maps, if there is a perfectoid space {X_\infty} such that {X_\infty \sim \lim_{\leftarrow i} X_i}, then {X_\infty} is unique up to unique isomorphism. In this scenario, I will often write {X_\infty = \lim_{\leftarrow i\in I} X_i} when such a perfectoid space exists.

Now, fix a smooth proper curve {\mathfrak{X}} over {\mathrm{Spec}(K)} of genus {g\geq 1}, with associated adic space {X= \mathfrak{X}^{\mathrm{ad}}} over {\mathrm{Spa}(K)}.

Lemma. There is a natural equivalence {\mathrm{F\acute{E}t}(\mathfrak{X}) \cong \mathrm{F\acute{E}t}(X)}.

Proof: This follows from Lutkebohmert’s {p}-adic Riemann existence theorem, or from a very cool theorem of Fresnel and Matignon: any irreducible, quasicompact, separated one-dimensional rigid analytic space is either affinoid or projective. \Box

Now fix a base point {x=\mathrm{Spa}(K) \in X}, and let {\pi_1(X,x)=\pi_1(\mathfrak{X},x)} denote the usual étale fundamental group from SGA; this is a profinite group depending only on {g} (up to isomorphism), and there is a natural fiber functor

\displaystyle F_x : \mathrm{F\acute{E}t}(X) \cong \pi_1(X,x)-\mathrm{Sets},

where {G}-Sets (for {G} a profinite group) denotes the category of finite sets with continuous (left) {G}-action. For any open subgroup {U < \pi_1(X,x)}, we have a corresponding connected finite étale cover {X_U \rightarrow X} (which is uniquely algebraizable to a covering {\mathfrak{X}_U \rightarrow \mathfrak{X}}).

Theorem. Suppose the Néron model of {\mathfrak{A}=\mathrm{Jac}(\mathfrak{X})} is an abelian scheme over {\mathrm{Spec}(\mathcal{O}_K)}. Then there is a natural perfectoid space {\tilde{X}} such that

\displaystyle \tilde{X} = \lim_{\substack{\leftarrow \\ U\rightarrow {1}}} X_U.

Under the natural map {q: \tilde{X} \rightarrow X}, {\tilde{X}} is naturally a {\pi_1(X,x)}-torsor over {X}, and any {Y \in \mathrm{F\acute{E}t}(X)} can be recovered from {F_x(Y)} as the categorical quotient {\tilde{X} \times_{\pi_1(X,x)}F_x(Y)}.

In the remainder of this post, I’ll sketch the construction of {\tilde{X}}. Here’s the idea: Let {A = \mathfrak{A}^{ad}} be the rigid analytic Jacobian of {X}, so our point {x} gives a natural closed immersion {X \rightarrow A}. On fundamental groups we get a natural surjection

\displaystyle \alpha: \pi_1(X,x) \twoheadrightarrow \pi_1(X,x)^{\mathrm{ab}} \cong \pi_1(A,e) \simeq \hat{\mathbf{Z}}^{2g}.

Set {U_n = \alpha^{-1}(p^n \hat{\mathbf{Z}}^{2g})}, and let {X_n := X_{U_n}} denote the associated covering of {X}; these form an inverse system in the obvious way. We are going to first construct a perfectoid space {X_\infty} such that {X_\infty = \lim_{\leftarrow n} X_n}, and then carefully pile all the remaining {X_U}‘s on top of {X_\infty}, using the almost purity theorem to ensure that this pileup remains perfectoid.

Let {A_n \in \mathrm{F\acute{E}t}(A)} denote a copy of {A} mapping to {A} under {[p^n]}, so {A_n \rightarrow A} is a finite étale covering of degree {p^{2gn}}. These form an inverse system in the obvious way.

Lemma. The map {X_n \rightarrow X} is obtained from the embedding {X \rightarrow A} by pulling back under {A_n \rightarrow A}.

Proof: Easy exercise. \Box

Lemma. There is a perfectoid space {A_\infty} such that {A_\infty = \lim_{\leftarrow n} A_n}.

Proof: This is Lemme A.16 in Pilloni-Stroh’s “Cohomologie coherent et representations Galoisiennes”. \Box

Lemma (“The pullback lemma”). Let {f:X \rightarrow Y} be a morphism of rigid analytic spaces, and let {Z \rightarrow Y} be a perfectoid space. If {f} is “blah”, where “blah” {\in} {\{}open immersion, closed immersion, finite étale, étale{\}}, then {Z \times_{Y} X} exists as a perfectoid space and {Z \times_{Y} X \rightarrow Z} is “blah”.

If {Z \sim \lim_{\leftarrow i} Y_i} for some cofiltered inverse system of rigid spaces {Y_i /Y} with qcqs transition maps, then {Z \times_{Y} X \sim \lim_{\leftarrow i}(Y_i \times_{Y}X)}.

Proof: It suffices to work locally on {X} and {Y}. We give a brief sketch:
Open immersion: Immediate.
Closed immersion: Pull back the ideal sheaf defining {X} in {Y} and argue as in § II.2 of Scholze’s torsion paper.
Finite étale: Immediate from the almost purity theorem.
Etale: Any such morphism is locally a composite of finite étale morphisms and open immersions, so we reduce to those cases. \Box

Lemma. The fiber product {X_\infty = A_\infty \times_{A} X} exists as a perfectoid space, and {X_\infty \sim \lim_{\leftarrow n} X_n}.

Proof: Immediate from the pullback lemma applied to the case where {Y=A}, {(Y_i)=(A_n)}, {Z=A_{\infty}}. \Box

Lemma. For any open subgroup {H < \pi_1(X,x)}, there is a natural perfectoid space {X_{H,\infty}} such that {X_{H,\infty} = \lim_{\leftarrow n}X_{H \cap U_n}}, and {X_{H,\infty}} is finite étale over {X_\infty}.

Proof: The space {X_{H,\infty}} is naturally a connected component of {X_{H} \times_{X} X_\infty}; by the pullback lemma, this fiber product is perfectoid and finite étale over {X_\infty}. The inclusion map {X_{H,\infty} \rightarrow X_{H} \times_{X} X_\infty} is finite étale, so we deduce that {X_{H,\infty} \rightarrow X_\infty} is finite étale. \Box

The spaces {(X_{H,\infty})_{H}} form a cofiltered inverse system of perfectoid spaces over {X_\infty}, with finite étale transition maps.

Lemma. The inverse limit

\displaystyle \lim_{\substack{\leftarrow \\ H \rightarrow {1}}} X_{H,\infty}

exists as a perfectoid space.

Proof: Argue locally on {X_\infty}: let {V \subset X_\infty} be an open affinoid perfectoid subset, with preimage {V_H \subset X_{H,\infty}}. By the definition of finite étale morphisms, each {V_H} is affinoid perfectoid; since the category of affinoid perfectoid spaces admits all limits, we get {\lim_{\leftarrow H} V_H} as an affinoid perfectoid space over {V}. \Box

Final lemma. Let {\tilde{X}} be the perfectoid space constructed in the previous lemma. Then {\tilde{X} \sim \lim_{U \rightarrow 1} X_U}.

Proof: Easy consequence of all the above. \Box

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